Samokhin, Alexander B.

福本, 康秀

Kyoto University Research Information Repository

Title非等方媒質中における電磁場の特異モード (非線形波動現象の数理と応用)Author(s)福本, 康秀; Samokhin, Alexander B.Citation数理解析研究所講究録 (2010), 1701: 35-39Issue Date2010-08URL http://hdl.handle.net/2433/169994RightType Departmental Bulletin PaperTextversionpublisherKyoto University(Yasuhide Fukumoto)Faculty of Mathematics, Kyushu UniversityMoscow State Inst. Radioeng. Alexander B. SamokhinMoscow State Institute of Radioengineering,Electronics and Automation (MIREA)1[5, 6]( )Vlasov ( ) [8, 3].[1] [2]2( ) $D$$E$ $H$ Maxwell :$rotE+\mu\frac=0$ , rot$H- \hat\frac=J$ ,$div\hat E=$ O, div$E=0$ . (2.1)1701 2010 35-39 35$J$ ( ) $\mu$ $\hat$:$\hat(x)=(\begin{array}{lll}\epsilon(x) \epsilon(x) \epsilon(x)\epsilon(x) \epsilon(x) \epsilon(x)\epsilon(x) \epsilon(x) \epsilon(x)\end{array})$ ; $\epsilon(x)=\epsilon(x)+i\sigma(x)/\omega(\in \mathbb{C})$ . (2.2)$\sigma$$\hat=\epsilon\hat$ ($\hat$) $\epsilon(x)$H\"older3 Maxwell (2.1)$\omega(\in \mathbb{R})$ $D$ $E,$ $H\propto$$e$$G(R):= \frac$ ; $R:=|x-y|$ , $k:=\omega\sqrt{}$ (23)$\hat(x)=\hat(x)/\epsilon$ (2.4)Maxwell (2.1) [7]:$E(x)+ \frac(\hat(x)-\hat)E(x)-$ pv. $\int[(\hat(y)-\hat)E(y)\cdot\nabla]\nabla G(R)dy$$-k \int(\hat(y)-\hat)E(y)G(R)dy=E(x)$ $(x\in D)$ . (2.5)p.v. $\int$(2.5)[7]3$\hat,$ $(2.5)$$\hat$(2.5)$\hat E=E-\hat((\hat-\hat)E)=E$ (3.1)$\sigma\neq 0$$\hat$$\hat$ $\hat$$\lambda(\in \mathbb{C})$$\hat-\lambda\hat$36$\lambda$$\hat-\lambda\hat$ Noether$\lambda=\beta\hat(x)\beta=\frac\epsilon(x)\beta\beta$ $(x\in D)$ (3.2)$\beta$$\hat=\epsilon(x)\hat$(3.2) $\lambda=\epsilon(x)=\epsilon(x)/\epsilon$ [1]$x$ ( $=\beta$ )$x$ $[2]$$\lambda$ (2.5)[1, 7]. $\beta$ $\beta$$D$ $\beta$ (3.2)Weyl ; $X$$\{\psi\in X\}$$||\psi||=1$ , $narrow\infty hm||\hat\psi-\lambda\psi||=0$ (3.3)$\lambda(\in \mathbb{C})$ [4](25) (3.1) (33)$\{\psi\in X\}$$\lambda$$\hat(x):=\frac$ (3.4)$\epsilon(0)qq=0$ (3.5)$q(|q|=1)$$S(\alpha)=\{$$\frac,$ $\alpha\gg\frac$ $(if ||\hat(0)||\neq 0)$$(if ||\hat(0)||=0))$(3.6)$q(|q|=1)$$S(\alpha)$ :$\varphi(q)=\{\begin{array}{l}1 (q\in S(\alpha))0(q\not\in S(\alpha)).\end{array}$ (3.7)373$\tilde(k)=\mathcal{F}[\psi(x)]=(2\pi)\int\psi(x)edx$ (3.8):$\tilde(k)=\gamma k\tilde(k)$ ,$\tilde(k)=\exp(-\frac-\frac)$ ; $q=k/|k|$ . (3.9)$\gamma(\in \mathbb{R})$$||E(x)||=1$ (3.10)$\gamma\frac[S(\alpha)\alpha\sqrt{}+(4\pi-S(\alpha))\beta\sqrt{}]=1$ (3.11)(3.9)$\alpha=\beta$ , $\betaarrow\infty$ (3.12)[1] Budko, N.V. and Samokhin, A. B. (2006) Spectrum of the volume integral operatorof electromagnetic scattering, SIAM J. Sci. Comput. 28, 682-700.[2] Budko, N.V. and Samokhin, A. B. (2007) Singular modes of the electromagnetic field,J. Phys. $A$ : Math. Theor. 40, 6239-6250.[3] Case, K. M. (1959) Singular modes of the electromagnetic field, Ann. Phys. 7, 349-364.[4] Hislop, P. D. and Sigal, I. M. (1996) Introduction to Spectml Theory: With Applica-tions to Schrodinger Operators (New York: Springer).[5] Hirota, M. and Fukumoto, Y. (2008) Energy of hydrodynamic and magnetohydrody-namic waves with point and continuous spectra, J. Math. Phys. 49, 083101.38[6] Hirota, M. and Fukumoto, Y. (2008) Action-angle variables for the continuous spec-trum of ideal magnetohydrodynamics, Phys. Plasmas 15, 122101.[7] Samokhin, R. (2009) Volume Integral Equation Method in Problems of MathematicalPhysics, COE Lecture Note 16, Faculty of Mathematics, Kyushu University.[8] Van Kampen, N. G. (1955) On the theory of stationary waves in plasmas, Physica21, 949-963.39

ヒトウホウ バイシツチュウ ニオケル デンジバ ノ トクイ モード ヒセンケイ ハドウ ゲンショウ ノ スウリ ト オウヨウ

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ヒトウホウバイシツチュウニオケルデンジバノトクイモードヒセンケイハドウゲンショウノスウリトオウヨウ

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ヒトウホウ バイシツチュウ ニオケル デンジバ ノ トクイ モード ヒセンケイ ハドウ ゲンショウ ノ スウリ ト オウヨウ

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Kyoto University Research Information Repository

ヒトウホウバイシツチュウニオケルデンジバノトクイモードヒセンケイハドウゲンショウノスウリトオウヨウ